Simplify the following expression and state the condition under which the simplification is valid. You can assume that $k \neq 0$. $p = \dfrac{2k^2 + 5k}{8k} \div \dfrac{4(2k + 5)}{-5} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $p = \dfrac{2k^2 + 5k}{8k} \times \dfrac{-5}{4(2k + 5)} $ When multiplying fractions, we multiply the numerators and the denominators. $p = \dfrac{ (2k^2 + 5k) \times -5 } { 8k \times 4(2k + 5) } $ $ p = \dfrac {-5 \times k(2k + 5)} {8k \times 4(2k + 5)} $ $ p = \dfrac{-5k(2k + 5)}{32k(2k + 5)} $ We can cancel the $2k + 5$ so long as $2k + 5 \neq 0$ Therefore $k \neq -\dfrac{5}{2}$ $p = \dfrac{-5k \cancel{(2k + 5})}{32k \cancel{(2k + 5)}} = -\dfrac{5k}{32k} = -\dfrac{5}{32} $